Normal-exponential-gamma distribution

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Parameters μR — mean (location)
k > 0\, shape
\theta > 0\, scale
Support x \in (-\infty, \infty)\!
PDF \propto \exp{\left(\frac{(x-\mu)^2}{4\theta^2}\right)}D_{-2k-1}\left(\frac{|x-\mu|}{\theta}\right)\,\!
Mean \mu
Median \mu
Mode \mu
Variance  \frac{\theta^2}{k-1} for  k>1
Skewness 0

In probability theory and statistics, the normal-exponential-gamma distribution (sometimes called the NEG distribution) is a three-parameter family of continuous probability distributions. It has a location parameter \mu, scale parameter \theta and a shape parameter k .

Probability density function[edit]

The probability density function (pdf) of the normal-exponential-gamma distribution is proportional to

f(x;\mu, k,\theta) \propto \exp{\left(\frac{(x-\mu)^2}{4\theta^2}\right)}D_{-2k-1}\left(\frac{|x-\mu|}{\theta}\right)\,\!,

where D is a parabolic cylinder function.[1]

As for the Laplace distribution, the pdf of the NEG distribution can be expressed as a mixture of normal distributions,

f(x;\mu, k,\theta)=\int_0^\infty\int_0^\infty\ \mathrm{N}(x| \mu, \sigma^2)\mathrm{Exp}(\sigma^2|\psi)\mathrm{Gamma}(\psi|k, 1/\theta^2) \, d\sigma^2 \, d\psi,

where, in this notation, the distribution-names should be interpreted as meaning the density functions of those distributions.


The distribution has heavy tails and a sharp peak[1] at  \mu and, because of this, it has applications in variable selection.